Blue Mage (青魔道士 [ao madoushi] in Japanese) is an oddball mage job and a relative newcomer to the series. It first appeared in FF5 and has made several appearances since. Blue Mages are recognizable by their blue attire, be that attire capes or tunics or other clothing. They usually wield curved sabers and/or rods. What makes them distinct is their system of learning abilities from monsters. Due to this specialty there aren't really any direct equivalent jobs, although in their nature of versatile/hybrid mages the jobs closest to it are the Red Mage and the Mystic Knight.
Despite the rarity of the job itself, there are quite a few examples of Blue Mages in the series. These are Strago Magus in FF6, Quistis Trepe in FF8, Quina Quen in FF9, Kimahri in FF10 and Raubahn in FF11.

Final Fantasy V

"Mages with the ability to learn monsters' special attacks"

Blue Mages are a versatile but restricted job. Versatile, because they have many types of spells that don't necessarily rely on their MAG stat alone, and restricted, because they do not have many options if you want to cast a certain type of spell.